Keeping in mind that \(u_i=0\)  in \(\omega _j\) , \(i\ne j\) ,
then \(\psi _a=\sum _{j=1}^{2s} (-1)^j u_j\)  is harmonic
in \(\dot{\Omega }_i\setminus \lbrace q_i\rbrace \) , \(i=1,...,r\) .
It is easy to see that
\(\dot{\Omega }_1\cup ...\cup \dot{\Omega }_r=D\setminus \lbrace q_1,...,q_r\rbrace \) .
Then the function \(\psi _a=\sum _{j=1}^{2s}(-1)^ju_j\)  is harmonic
in \(D\setminus \lbrace q_1,...,q_r\rbrace \)  (see [1]}),
moreover \(\psi _a\)  is bounded so, by Schwarz's removable
singularity principle (see [2]})
\(\psi _a\)  is harmonic in \(D\)  and, by construction,
\(U=|\psi _a|\)  in \(D\)  and \(U=|\Phi ^a|=\Phi \)  on \(\partial D\) .
